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Chapter 3: Problem 15

For each quadratic function, (a) write the function in the form \(P(x)=a(x-h)^{2}+k,\) (b) give the vertex of the parabola, and (c) graph the function. Do not use a calculator. $$P(x)=3 x^{2}+4 x-1$$

Short Answer

Expert verified
Vertex form: \(P(x) = 3(x + \frac{2}{3})^2 - \frac{7}{3}\); Vertex: \((-\frac{2}{3}, -\frac{7}{3})\).

Step by step solution

01

Identify Coefficients

The given quadratic equation is in the form $ax^2 + bx + c$. Here, identify the values of $a$, $b$, and $c$ from the equation $P(x) = 3x^2 + 4x - 1$. We have $a=3$, $b=4$, and $c=-1$.
02

Completing the Square

We need to express the function in the vertex form \(P(x)=a(x-h)^2+k\). Start by factoring out 3 from the first two terms: \(P(x) = 3(x^2 + \frac{4}{3}x) - 1\). Now, complete the square for the expression in the parenthesis: take \((\frac{4}{3} \div 2)^2 = (\frac{2}{3})^2 = \frac{4}{9}\) and add and subtract this inside the parenthesis: \(P(x) = 3(x^2 + \frac{4}{3}x + \frac{4}{9} - \frac{4}{9}) - 1\). This becomes \(P(x) = 3((x + \frac{2}{3})^2 - \frac{4}{9}) - 1\).
03

Simplify the Expression

Distribute the 3 back through the completed square expression and simplify: \(P(x) = 3(x + \frac{2}{3})^2 - \frac{12}{9} - 1\). Combine like terms: \(P(x) = 3(x + \frac{2}{3})^2 - \frac{12}{9} - \frac{9}{9} = 3(x + \frac{2}{3})^2 - \frac{21}{9}\). Further simplify the constant term: \(P(x) = 3(x + \frac{2}{3})^2 - \frac{7}{3}\).
04

Determine the Vertex

The vertex form of the equation is \(P(x) = 3(x + \frac{2}{3})^2 - \frac{7}{3}\), which is in the form \(P(x) = a(x-h)^2+k\). Here, \(h = -\frac{2}{3}\) and \(k = -\frac{7}{3}\). Thus, the vertex of the parabola is \((-\frac{2}{3}, -\frac{7}{3})\).
05

Sketch the Graph

Plot the vertex \((-\frac{2}{3}, -\frac{7}{3})\). Since \(a = 3\), the parabola opens upwards and is narrower than the standard parabola (as \(a > 1\)). Sketch the parabola ensuring it passes through the vertex and has the general upward shape expected for a positive leading coefficient.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertex Form
The vertex form of a quadratic function is a useful way to express the function and easily identify its vertex. It's written as \[ P(x) = a(x-h)^2 + k \]. Here, \(a\) affects the width and direction of the parabola, while \((h, k)\) is the vertex of the parabola.
  • The value \(a\) indicates whether the parabola opens upwards (if \(a > 0\)) or downwards (if \(a < 0\)).
  • The vertex \((h, k)\) provides a key point on the graph and is the 'turning point' of the parabola.

Using the example \(P(x) = 3(x + \frac{2}{3})^2 - \frac{7}{3}\), we easily find that the vertex is \((-\frac{2}{3}, -\frac{7}{3})\). This form makes graphing intuitive, as you can directly plot the vertex and then use \(a\) to determine the parabola's shape.
Completing the Square
Completing the square is a method to transform a quadratic equation into vertex form. This involves creating a perfect square trinomial from a quadratic, which can then be rearranged into vertex form, making it simpler to identify the vertex.
Here's a simple breakdown of the process:
  • Factor out the coefficient \(a\) from the quadratic and linear terms.
  • Find the term needed to complete the square for the expression \(x^2 + \frac{b}{a}x\). This is found by taking \(\left(\frac{b}{2a}\right)^2\).
  • Add and subtract this term inside the parentheses, then rearrange to form a square trinomial plus or minus some constant.

In our example:- Start with \(3(x^2 + \frac{4}{3}x)\), - Add and subtract \(\frac{4}{9}\), turning it into a perfect square: \(3((x + \frac{2}{3})^2 - \frac{4}{9})\).- Distribute back, and simplify to get \(P(x) = 3(x + \frac{2}{3})^2 - \frac{7}{3}\). This transformed equation makes further analysis easier.
Graphing Parabolas
Graphing parabolas becomes straightforward when using the vertex form of a quadratic function because you can directly locate the vertex. Here's how to sketch the graph:Follow these steps:
  • Identify the vertex \((h, k)\) from the vertex form \(P(x) = a(x-h)^2 + k\). The vertex in our case is \((-\frac{2}{3}, -\frac{7}{3})\).
  • Since \(a = 3\), the parabola opens upwards and is narrower than a standard parabola. This is because any \(a > 1\) narrows the graph.
  • Sketch the parabola beginning at the vertex. The axis of symmetry goes through this point, which helps ensure a balanced graph.
  • Use other points from the function, if needed, to ensure accuracy in the shape and direction of the curve.

The vertex provides a clear reference, and combining this with the leading coefficient \(a\) offers a vivid idea of the parabola's orientation and width.

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Most popular questions from this chapter

Solve each quadratic equation by completing the square. $$x^{2}-2 x=2$$

Evaluate the discriminant, and use it to determine the number of real solutions of the equation. If the equation does have real solutions, tell whether they are rational or irrational. Do not actually solve the equation. $$2 x^{2}-4 x+1=0$$

Solve each equation. For equations with real solutions, support your answers graphically. $$4 x^{2}-12 x=-11$$

Solve each equation. For equations with real solutions, support your answers graphically. $$-3 x^{2}+4 x+4=0$$

Solve each inequality analytically. Support your answers graphically. Give exact values for endpoints. (a) \(x^{2}+5 x<2\) (b) \(x^{2}+5 x \geq 2\)
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